Let $X$ be a finite connected pointed CW-complex and $H_{\ast}(\Omega X)$ the integral homology of the loop space on $X$. Are the homology groups $H_{n}(\Omega X)$ finitely generated abelian groups for any $n$ ?
If the answer is negative, what are the sufficient conditions to impose on $\pi_{1}(X)$ such that the homology groups $H_{n}(\Omega X)$ turns out to be finitely generated ?
My goal is to collect different sufficient conditions on the fundamental group for which a positive answer holds.
This is true for finite $\pi_1$ and false for infinite $\pi_1$: Let $\widetilde{X}$ denote the universal cover of $X$, then $\Omega\widetilde{X}$ is the unit connected component of $\Omega X$, and $\Omega X = \coprod_{\pi_1(X)} \Omega\widetilde{X}$. So if $\pi_1$ is infinite, then certainly $H_0(\Omega X)$ is not finitely generated as others have noted in the comments, and indeed if $\Omega\widetilde{X}$ has any nontrivial homology group (which is true unless $\widetilde{X}$ is contractible), some higher homology group of $\Omega X$ will be an infinite direct sum of nontrivial abelian groups, so also not finitely generated.
If $\pi_1$ is finite, on the other hand, $\widetilde{X}$ is again a finite CW complex, so in that case it suffices to look at the simply-connected case. For a simply-connected finite CW-complex $X$, $H_*(\Omega X)$ indeed consists of finitely generated abelian groups, which goes back to Serre (and is proved easily using the spectral sequence named after him).
